Stochastic Normalizing Flows for Inverse Problems: A Markov Chains Viewpoint

To overcome topological constraints and improve the expressiveness of normalizing ow architectures, Wu, Kohler, and Noe introduced stochastic normalizing ows which combine deterministic, learnable ow transformations with stochastic sampling methods. In this paper, we consider stochastic normalizing ows from a Markov chain point of view. In particular, we replace transition densities by general Markov kernels and establish proofs via Radon-Nikodym derivatives, which allows us to incorporate distributions without densities in a sound way. Further, we generalize the results for sampling from posterior distributions as required in inverse problems. The performance of the proposed conditional stochastic normalizing ow is demonstrated by numerical examples.